# Geometric description of discrete power function associated with the sixth Painlev\'e equation

## TL;DR

This paper explores the geometric structure of a discrete power function linked to the sixth Painlevé equation, revealing its embedding in a symmetric lattice and clarifying the odd-even pattern in related formulas.

## Contribution

It demonstrates the embedding of the discrete power function in a cubic lattice with specific symmetry and relates the symmetry groups of the Painlevé equation to the lattice structure.

## Key findings

- The discrete power function is embedded in a cubic lattice with tenilde;W(3A_1^{(1)}) symmetry.
- The symmetry group tenilde;W(D_4^{(1)}) of P_VI is related to the lattice symmetry group.
- The odd-even structure in explicit formulas is explained via translations in the symmetry group.

## Abstract

In this paper, we consider the discrete power function associated with the sixth Painlev\'e equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $\widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $\widetilde{W}(3A_1^{(1)})$ as a subgroup of $\widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{\rm VI}$, we show how to relate $\widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $\widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $\tau$ function.